P.-G. Pripoae and C.-L. Pripoae
Abstract. In this paper, we study the sets of the left invariant and of the bi-invariant connections on Lie groups, endowed with some additional properties: symmetry, flatness, Ricci-flatness, etc. Moreover, we give some new examples in low dimensions for some special types of affine connec- tions.
M.S.C. 2010: 53B05, 53B20, 22E15, 22E60.
Key words: affine differential manifolds; Lie groups; left-invariant connections; bi- invariant connections; flat connections; Ricci-flat connections; Ricci-symmetric con- nections; Cartan connections; mixed flat connections.
1 Introduction
On Lie groups, the invariant geometries are an important tool for testing conjectures and for classifying different differential/affine/metric objects. In particular, affine connections that are left or bi-invariant with respect to translations were considered in many papers ([8], [7], [1], [14],[15], [16], etc).For compact Lie groups, the set of bi- invariant connections was classified by Laquer ([5],[6]). We are not aware of a similar result, in the non-compact case.
For a n-dimensional Lie group G, the left-invariant connections are completely modelled as (1,2)-tensors on the Lie algebra L(G), thus their set may be identified withRn3. When additional properties are considered, this set reduces; our aim is to determine how and why.
There exist similar studies for specific families of affine connections on differen- tiable manifolds, but the techniques and results are of a completely different nature ([2], [3],[4]).
In this paper, we study (some sub-) sets of invariant connections and to what extent they may classify the Lie algebras or the Lie groups. In §2 we determine the sets of symmetric left-invariant connections, of the flat connections and of the symmetric and flat connections, respectively; examples are given in dimensions 2 and 3; we suggest two new conjectures.
Balkan Journal of Geometry and Its Applications, Vol.24, No.1, 2019, pp. 51-64.∗
⃝c Balkan Society of Geometers, Geometry Balkan Press 2019.
In§3 we define the mixed flat affine differential manifolds and we give examples of such (left-invariants) structures on Lie groups of low dimensions; the set of all the mixed flat left-invariant connections is determined in the general case.
In§4 we characterize the sets of left-invariant connections which are: Ricci-flat, symmetric and Ricci-flat, Ricci-symmetric, symmetric and Ricci-symmetric respec- tively.
In§5 we determine the sets of bi-invariant connections on the 2-dimensional non- commutative Lie group, which are symmetric or flat; there exists a unique symmetric and flat connection. On the Heisenberg group, we find the sets of all the bi-invariant connections, as well as the subset of symmetric ones.
2 The setting
Consider G a n-dimensional Lie group and L(G) its Lie algebra. For each a ∈ G, we denote byLa and Ra the left and right translations onG associated toa, given by La(x) = ax and Ra(x) = xa, for all x ∈ G. An affine connection ∇ on G is left-invariant if, for any vector fieldsX andY onG,
∇(La)∗X(La)∗Y = (La)∗∇XY.
The right-invariant connections are defined in a similar manner. A connection is called bi-invariant if it is simultaneously left and right-invariant.
We denote by C(G), C(G)l, C(G)sl, C(G)b, C(G)sb the sets of (affine) connec- tions, of left-invariant, of symmetric left-invariant, of bi-invariant and of symmetric bi-invariant ones.
Let fix a basis {E1, ..., En}ofL(G). Each∇ ∈ C(G)lmay be writen as∇EiEj = ΓkijEk, for everyi, j= 1, n, with real coefficients Γkij.
The set C(G)l is in one-to-one correspondence with the set of (1,2)-tensor fields onL(G), so may be identified with the real vector space Rn3. All the connections studied hereafter will belong to this ”ambient space”.
Proposition 2.1. The set C(G)sl is an affine subspace in C(G)l, of affine dimen- sionn2(n+ 1)/2. Moreover, C(G)sl is a linear subspace in C(G)l if and only ifG is commutative.
Proof. Consider a basis{E1, ..., En}ofL(G) with structural constantsckij, fori, j, k= 1, n. Write∇ ∈ C(G)sl as∇EiEj = ΓkijEk, for everyi, j= 1, n, with real coefficients Γkij, such that
(2.1) Γkij−Γkji−ckij = 0.
We have here a system ofn3affine equations in the unknowns Γkij. It follows that the coefficients Γkij parameterizeC(G)sl as an affine subspace inRn3, of affine dimension n2(n+ 1)/2. The second property is obvious, as allckij vanish. We point out also that these two properties do not depend on the choice of the basis inL(G).
Proposition 2.2. (i)The set of flat connections inC(G)l is the (non-void) intersec- tion ofn2(n2−1)/3hyperquadrics inRn3, forn≥1. Moreover, all these hyperquadrics have a center in the origin if and only ifGis commutative.
(ii) The set of local Euclidean (i.e. symmetric and flat) connections in C(G)sl is the intersection ofn2(n2−1)/3 hyperquadrics withn2(n−1)/2affine hyperplanes in Rn3, forn≥1. Moreover, this set contains the origin if and only ifGis commutative.
Proof. (i) Fix a basis{E1, ..., En}ofL(G); denote bycijk the structural constants, by Γkij the coefficients of an arbitrary flat left invariant connection onGand byRijkl the components of its curvature tensor fieldR. The vanishing ofRyields to
(2.2) ΓskiΓljs−ΓsjiΓlks−csjkΓlsi= 0,
for alli, j, k, l= 1, n. This system of (apparently)n4quadratic equations depends on then3unknowns Γijk; in fact, onlyn2(n2−1)/3 equations are effective, because exactly n2(n2−1)/3 coefficientsRijkl are independent, and the others may be deduced from them (see, for example [18]). Each equation in (2.2) defines an affine hyperquadric in Rn3; the quadratic part is independent ofG(depends only onn), but the linear part depends onG, through the structural constants.
The linear partcsjkΓlsivanishes, for alli, j, k, l= 1, nand for all coefficients of the connections if and only if all the structural constants vanish.
The system (2.2) is compatible, as it admits the trivial solution (which corresponds to the Cartan-Schouten connection∇−).
(ii) From (2.1) we deduce the condition involving then2(n−1)/2 affine hyperplanes in Rn3, as well as the characterization concerning the case when all the structural constants vanish.
Remark 2.3. Supposen= 2 andGcommutative.
(i) We may reduce the parameterizing set of flat connections in C(G)l to the set S of solutions of the (”minimal”) system of equations
R1112= 0 , R1212= 0 , R2112= 0 , R2212= 0.
In other words,
Γ221Γ112−Γ211Γ122= 0
Γ122Γ111+ Γ222Γ112−Γ112Γ121−Γ212Γ122= 0
Γ212Γ221+ Γ121Γ211−Γ111Γ221−Γ211Γ222= 0 Γ122Γ211−Γ112Γ221= 0, (redundant)
in the eight unknowns Γ111,Γ112,Γ121,Γ122,Γ211,Γ212,Γ221,Γ222.
For simplicity, we denote the variables x1 = Γ111, x2 = Γ112, x3 = Γ122, x4 = Γ211, x5= Γ212, x6 = Γ222,x7= Γ121, x8= Γ221. The set of flat left invariant connections in Gis parameterized by the setS of the solutions of the following system of quadratic equations inR8:
x8x2−x4x3= 0 x3x1+x6x2−x2x7−x5x3= 0 x1x8+x6x4−x5x8−x7x4= 0.
An elementary calculation determinesS, as the union of the following submanifolds inR8, of dimension 6,5,5 and 4, respectively:
{(x1, x2, x8x2(x4)−1, x4, x5, x7+x8(x5−x1)(x4)−1, x7, x8)|x1, x2, x4, x5, x7, x8∈R, x4̸= 0} {(x1, x2, x3,0, x5, x7+x3(x5−x1)(x2)−1, x7,0)|x1, x2, x3, x5, x7∈R, x2̸= 0}
{(x1,0, x3,0, x1, x6, x7, x8)|x1, x3, x6, x7, x8∈R} (a 5−plane) {(x1,0,0,0, x5, x6, x7,0)|x1, x5, x6, x7∈R} (a 4−plane)
(ii) We may reduce the parameterizing set of symmetric flat connections inC(G)l
to the set of solutions of the (”minimal”) system of equations
R1121 = 0 , R2121 = 0 , R1122 = 0 , R2122 = 0 , Γ112= Γ121 , Γ212= Γ221. In other words
Γ212Γ112−Γ211Γ122= 0
Γ122Γ111+ Γ222Γ112−(Γ112)2−Γ212Γ122= 0 Γ111Γ212+ Γ211Γ222−(Γ212)2−Γ112Γ211= 0
Γ122Γ211−Γ112Γ212= 0 (redundant)
in the six unknowns Γ111,Γ112,Γ122,Γ211,Γ212,Γ222.
For simplicity, we denote the variables x1 = Γ111, x2 = Γ112, x3 = Γ122, x4 = Γ211, x5 = Γ212, x6 = Γ222. The set of symmetric flat left invariant connections in G is parameterized by the set S of the solutions of the following system of quadratic equations inR6:
x5x2−x4x3= 0 x3x1+x6x2−(x2)2−x5x3= 0 x5x1+x6x4−(x5)2−x2x4= 0
An elementary calculation determinesS, as the union of the following submanifolds inR6, of dimension 4,3,3 and 2, respectively:
{(x1, x2, x5x2(x4)−1, x4, x5, x2+x5(x5−x1)(x4)−1)|x1, x2, x4, x5∈R, x4̸= 0} {(x1, x2, x3,0,0, x2−x3x1(x2)−1)|x1, x2, x3∈R, x2̸= 0}
{(x1,0, x3,0, x1, x6)|x1, x3, x6∈R} (a 3−plane) {(x1,0,0,0,0, x6)|x1, x6∈R} (a 2−plane)
(Another method consists in particularizingx7:=x2and x8:=x5 in (i).) Remark 2.4. Suppose n = 2 and G non-commutative. We may choose a basis {E1, E2} ofL(G) such that [E1, E2] =E1.
(i) We may reduce the parameterizing set of flat connections in C(G)l to the set of solutions of the (”minimal”) system of equations
R1112= 0 , R1212= 0 , R2112= 0 , R2212= 0 In other words
Γ221Γ112−Γ211Γ122−Γ111= 0
Γ122Γ111+ Γ222Γ112−Γ112Γ121−Γ212Γ122−Γ112= 0
Γ212Γ221+ Γ121Γ211−Γ111Γ221−Γ211Γ222−Γ211= 0
Γ122Γ211−Γ112Γ221−Γ212= 0
in the eight unknowns Γ111,Γ112,Γ121,Γ122,Γ211,Γ212,Γ221,Γ222.
For simplicity, we denote the variables x1 = Γ111, x2 = Γ112, x3 = Γ122, x4 = Γ211, x5= Γ212, x6 = Γ222,x7= Γ121, x8= Γ221. The set of flat left invariant connections in Gis parameterized by the setS of the solutions of the following system of quadratic equations inR8:
x8x2−x4x3−x1= 0 x3x1+x6x2−x2x7−x5x3−x2= 0 x1x8+x6x4−x5x8−x7x4+x4= 0
x4x3−x8x2−x5= 0.
We obtainSas the union of the following submanifolds inR8, of dimension 4,4,3 and 3 respectively:
{(±√
−x2x4, x2,∓x2(x6−x7−1) 2√
−x2x4 , x4,∓√
−x2x4, x6,
x7,∓x4(x6−x7+ 1) 2√
−x2x4 )|x2, x4, x6, x7∈R, x2x4<0} {(0,0, x3,0,0, x6, x7, x8)|x3, x6, x7, x8∈R}
{(0, x2, x3,0,0, x7+ 1, x7,0)|x2, x3, x7∈R , x2̸= 0} {(0,0,0, x4,0, x6, x6+ 1, x8)|x4, x6, x8∈R.}
(ii) We want to parameterize the set of symmetric flat connections inC(G)l, so we suppose, in addition to (i), that (1.1) holds. As the only non-null structural constants arec112=−c121= 1, we deduce
Γ112−Γ121−1 = 0 , Γ212= Γ221.
We obtainSas the union of the following submanifolds inR8, of dimension 2,2,1 and 1 respectively:
{(±√
−x2x4, x2,±x2(x2+ 1)
√−x2x4 , x4,∓√
−x2x4,−x2−2, x2−1,∓√
−x2x4)|x2, x4∈R, x2x4<0} {(0,0, x3,0,0, x6,−1,0)|x3, x6∈R}
{(0,−1, x3,0,0,−1,−2,0)|x3∈R}
{(0,0,0, x4,0,−2,−1,0)|x4∈R}. Remark 2.5. Supposen= 3 andGcommutative.
(i) We may reduce the parameterizing set of flat connections in C(G)l to the set S of solutions of the (”minimal”) system of 24 equations
Ri112= 0 , Ri113= 0 , Ri123= 0 , Ri212= 0 Ri213= 0 , Ri223= 0 , Ri313= 0 , Ri323= 0 fori∈ {1,2,3}in the 27 unknowns Γijk. In other words
Γ111Γi21+ Γ211Γi22+ Γ311Γi23−Γ121Γi11−Γ221Γi12−Γ321Γi13= 0,
fori∈ {1,2,3}and another 21 similar equations (each of them defining a quadratic variety inR27). The system is compatible, as it admits the trivial solution∇−, with all the coefficients null.
(ii) We may now reduce the parameterizing set of symmetric flat connections in C(G)l, by adding the 9 equations Γijk = Γikj for i, j, k ∈ {1,2,3}, j < k, to the system from the previous remark. This new system is also compatible, as it admits the solution∇−.
Remark 2.6. Suppose n = 3 and Gnon-commutative. The study becomes more complex(and, due to the lack of space, will be carried out elsewhere), as we must take into account the classification of the 3-dimensional Lie algebras ([8], [12], [18]),for the following non-commutative Lie groups: the Heisenberg group H3,the orthogo- nal group O(3), the special linear group SL(2,R), the Lorentz group O(1,2), the Euclidean motions groupE(2) and the Minkowski motions group E(1,1).
We sketch here the case of the Heisenberg group H3. Fix a basis{E1, E2, E3}of L(H3) such that [E1, E2] =E3.
The set of flat connections in C(H3)l is given by a system of 24 equations in the 27 unknowns Γijk.
For the set of symmetric and flat connections inC(H3)l, we must add the following 8 equations
Γ312−Γ312−1 = 0, Γ112= Γ121, Γ113= Γ131, Γ123= Γ132 Γ212= Γ221, Γ213= Γ231, Γ223= Γ232, Γ312= Γ321, Γ313= Γ331, and we obtain a system of 33 equations in the 27 unknowns Γijk.
The set of Ricci-flat connections in C(H3)l is given by a system of 9 equations in the 27 unknowns Γijk; for the symmetric and Ricci-flat connections we obtain a system of 18 equations.
The set of symmetric and Ricci-flat connections inC(H3)lis given by a system of 9 equations in the 27 unknowns Γijk; for the symmetric and Ricci-flat connections we obtain a system of 18 equations.
The set of Ricci-symmetric connections inC(H3)lis given by a system of 3 equa- tions in the 27 unknowns Γijk; for the symmetric and Ricci-symmetric connections we obtain a system of 12 equations.
The set of symmetric and Ricci-symmetric connections in C(H3)l is given by a system of 3 equations in the 27 unknowns Γijk; for the symmetric and Ricci-symmetric connections we obtain a system of 12 equations.
Remark 2.7. Supposen≥4. The system (2.2) admits always a non-trivial solution (hence a line of solutions), for example for the Cartan-Schouten connection∇0, which is half of the Lie bracket onL(G).
(i) Instead, it is not evident at all if the system (2.1)+(2.2) admits always a solution, as for growing n it becomes overdetermined ( with n2(n−1)(2n+ 5)/6 equations vs. n3unknowns). This is an important topic, which gave rise to the study of the left structures (affine structures) and suggested, for example, the Auslander- Milnor’s conjecture ([9]):”On each solvable Lie group there exists a symmetric and flat left-invariant connection”. This conjecture was refuted through counterexamples constructed on filiform algebras. We may weaken the Auslander-Milnor conjecture, through the following two conjectures:
Conjecture 1. On each Lie group there exists a symmetric and Ricci-flat left- invariant connection.
Conjecture 2. On each Lie group there exists a symmetric and Ricci-symmetric left-invariant connection.
Obviously, Conjecture 1 true implies Conjecture 2 true. (For the Conjecture 1, the 2-dimensional case is not interesting, as (in this case) every Ricci-flat connection is flat.)
(iii) Another interesting problem is to determine all the non-flat left-invariant connections which are Ricci-flat. The Riemannian case is solved (there are none !); in the indefinite case, there exist such connections. In the following we shall deal with the pure affine case.
Remark 2.8. A left-invariant connection ∇ ∈ C(G)l is called Cartan connection ([13]) if∇XX = 0 for anyX ∈L(G). It is easy to prove that any such connection is of the form
(2.3) ∇XY =1
2([X, Y] +T(X, Y]),
for anyX, Y ∈L(G) and any skew-symmetric tensor fieldTof type (1,2) onL(G). (It follows that extendingT gives precisely the torsion field of∇.) As examples we have the classical Cartan-Schouten connections∇−, ∇+, ∇0 and any other one collinear with them. Obviously, we have: (i) the set of left-invariant Cartan connections may be parameterized byRn2 (n−1)2 , wheren=dimG; (ii) there exists a unique symmetric left-invariant Cartan connection, namely∇0 (in fact it is even bi-invariant);(iii) the Propositions 2.1 and 2.2,(i) may be re-written accordingly; (iv) as a Cartan connection satisfies∇XY+∇YX = 0, for everyX, Y ∈L(G), we see that such connection is the skew-symmetric analogue of symmetric (i.e. torsion-free) one.
3 Mixed flat connections
LetM be an-dimensional differentiable manifold and∇a linear (affine) connection onM. Denote R,RicandT the curvature, the Ricci and the torsion tensor fields of
∇, respectively. We define a (1,3)-tensor field onM, byU(X, Y)Z :=R(X, Y)Z− (n−1)−1{Ric(Y, Z)X−Ric(X, Z)Y}. We callU the Riemann-Ricci tensor field; it is skew-symmetric in the first two variables and traceless in the third variable.
Definition 3.1. The affine differentiable manifold (M,∇) is called mixed flat ifU identically vanishes.
In the following, we make some comments and give some examples. More details about the geometry of mixed flat manifolds as well as their applications in the Theory of Relativity will appear elsewhere ([15]).
Remark 3.2. (i) Obviously, every flat affine connection on a differentiable manifold is mixed flat; every mixed flat and Ricci flat affine connection must be flat.
(ii) In dimension 2, any affine differentiable manifold (M,∇) is mixed flat.
(iii) Consider a mixed flat affine differentiable manifold (M,∇). If T = 0, then Ricis symmetric and cyclic-parallel.
(iv) Let (N, g) be a semi-Riemannian manifold of dimension greater than 3 and
∇ its Levi-Civita connection. Then (N,∇) is mixed flat if and only if (N, g) has constant sectional curvature. It follows that this new notion is irrelevant for the semi- Riemannian geometry. Nonetheless, the notion of mixed flatness is important in the non-Riemannian case, as it extends the notion of constant sectional curvature beyond the frontiers of metric theories.
(v) LetGbe a n-dimensional Lie group and ∇ a left-invariant connection onG.
Altogether withRandRic, the tensor field U is also left-invariant.
The study of mixed flat connections on Lie groups is particularly relevant, because the examples of Lie groups, which admit left-invariant Riemannian metrics with con- stant sectional curvature, are quite rare (cf. [8]).
Consider the Cartan-Schouten connections∇−,∇+,∇0. We remark that∇−and
∇+ are always flat, hence they are also mixed flat. The connection∇0 is mixed flat if and only if
(3.1) (n−1)[[X, Y], Z] =B(Y, Z)X−B(X, Z)Y,
for everyX, Y, Z ∈L(G), whereB is the Killing form onG. Of course, this relation holds for every Levi-Civita connection of a bi-invariant metric with constant sectional curvature, as previously pointed out.
In the sequel we provide an example when this condition holds, in the affine differential setting.
Proposition 3.3. (i) The set of mixed flat connections in C(G)l is the (non-void) intersection of n2(n2 −1)/3 hyperquadrics in Rn3, for n ≥ 1. Moreover, if G is commutative, then all these hyperquadrics have a center in the origin.
The proof is similar to that of Proposition 2.2.
4 On the symmetry of the Ricci tensor
LetM be an-dimensional differentiable manifold and∇a linear (affine) connection onM. Denote R,RicandT the curvature, the Ricci and the torsion tensor fields of
∇, respectively. In the affine differentiable setting, the symmetry of the Ricci tensor is a quite subtle property. It is known ([11]) that, for a symmetric connection ∇, the Ricci-symmetry is equivalent with the local existence of a volume form which is∇-parallel (i.e. ∇ is locally equiaffine). We shall extend this result for arbitrary connections.
We define a 2-formtonM, by t(Y, Z) =trace{X→∑
[T(T(X, Y), Z) + (∇XT)(Y, Z)]}, with cyclic sum afterX, Y, Z∈ X(M).
Remark 4.1.(i) We distinguish the special cases when the 2-formtis exact or closed.
Each such property defines a new interesting family of affine differential manifolds.
(ii) Suppose there exists a one-formαonM such thatT(Y, Z) =α(Z)Y−α(Y)Z (i.e. ∇ is semi-symmetric). Then
t(Y, Z) = (∇Yα)Z−(∇Yα)Z andt= 0 if and only ifαis closed.
From the first Bianchi identity, by contracting, we get
Ric(Y, Z)−Ric(Z, Y) =−traceR(Y, Z) +t(Y, Z).
We get the following
Lemma 4.2. The Ricci tensor is symmetric if and only if traceR(Y, Z) =t(Y, Z), for every vector fieldsY, Z.
Letω be a local volume element onM. Then, there exists a one-formτ such that
∇Yω=τ(Y)ω (i.e. ω is a∇-recurrentn-form, with recurrency factorτ). One knows (cf. [11]) that
R(Y, Z)ω=−[traceR(Y, Z)]ω.
On another hand, we derive
R(Y, Z)ω= 2[dτ(Y, Z)]ω+τ(T(Y, Z))ω.
The last two relations lead to
(4.1) traceR(Y, Z) + 2[dτ(Y, Z)] +τ(T(Y, Z)) = 0.
Theorem 4.3. Let ∇ be an affine connection on the differentiable manifoldM, with t= 0.
(i) If there exists a local ∇-parallel volume element, thenRic is symmetric.
(ii) SupposeRic is symmetric and suppose there exists a (local) volume element with recurrency factorτ such that ImT ⊂kerτ. Then there exists a local ∇-parallel volume element onM.
Proof. From Lemma 4.2,Ricis symmetric if and only iftraceR(Y, Z) = 0 for every vector fieldsY, Z.
(i) Let ω be a local ∇-parallel volume element on M, so its recurrency factor τ = 0. From (4.1) it follows that traceR(Y, Z) = 0, which proves the symmetry of Ric.
(ii) Suppose Ric is symmetric and ω is the volume element with the required property. It follows thattraceR(Y, Z) = 0, for every vector fieldsY, Z. Relation (4.1) implies that
2[dτ(Y, Z)] +τ(T(Y, Z)) = 0.
From the hypothesis,τ(T(Y, Z)) = 0, for every vector fields Y, Z. It follows thatτ is an exact one-form; thus there exists a functionf such thatτ =−dlnf. The volume elementf ωis ∇-parallel.
Remark 4.4. (i) The theorem 4.3.,(ii) generalizes the quoted result from [11], which can be recovered forT = 0.
(ii) The generalization is effective, as may be seen from the following example.
Let consider a 2-dimensional non-commutative Lie group as in the Remark 2.4 with the formulas therein. Let∇be the left-invariant connection with all the components null, except Γ111= 1 and Γ112=−1. ThenRic12=Ric21(hence Ric is symmetric);∇ is not flat nor Ricci-flat, asR1112=Ric12=−1. The connection∇ is non-symmetric, asT121 =−2.
A short calculation proves that t = 0, so there exist non-trivial cases when the hypothesis in theorem 4.3.,(ii) applies.
(Here the connection∇ is semi-symmetric, as in Remark 4.1., for the closed left- invariant one-formαsuch thatα(E2) =−2 andα(E1) = 0).
(iii) For any Lie groupG, the Cartan-Schouten connection∇− has the following properties: is bi-invariant;T =−[,] onL(G);t= 0. This is in agreement with theo- rem 4.3., as∇− is Ricci-flat (hence Ricci-symmetric) and each left-invariant volume element is∇−-parallel.
This example shows that the Conjecture 1 is true if we replace the requirement T = 0 witht= 0.
5 The set of bi-invariant connections
LetGbe a n-dimensional Lie group. The following (known) result characterizes the bi-invariant connections (see for example [13] for a partial sketch of proof).
Theorem 5.1. For a left-invariant connection ∇ ∈ Cl(G) the following claims are equivalent:
(i)∇is bi-invariant;
(ii)for everyX, Y, Z∈L(G)we have
(5.1) [X,∇YZ]− ∇[X,Y]Z− ∇Y[X, Z] = 0;
(iii)∇ isad-invariant, i.e., for every X ∈L(G)we haveadX∇= 0;
(iv)∇isAd-invariant, i.e., for everyX, Y ∈L(G)anda∈G, we have∇AdaXAdaY = Ada(∇XY).
Remark 5.2. (i) It is well known that the setC(G)bis always non-void, as it contains the Cartan-Schouten connections∇−, ∇+ and ∇0. (On L(G), they act as the null operator, as the Lie bracket and half of the Lie bracket, respectively). The connection
∇0is symmetric, soC(G)sbis also non-void. The connection∇−is flat, but in general it is not symmetric.
(ii) Using (2.3), we see that a (general) Cartan connection is bi-invariant if and only if the skew-symmetric tensor fieldTsatisfies [X, T(Y, Z)]−T([X, Y], Z)−T(Y,[X, Z]) = 0 for everyX, Y, Z ∈L(G) (i.e. T isad-invariant). A non-trivial example is the fol- lowing: considerωa left-invariant one-form onG, vanishing on the derived algebra of L(G) (i.e. ω([X, Y]) = 0 for everyX, Y ∈L(G)).DefineT(X, Y) :=ω(X)Y−ω(Y)X.
(iii) In the following, we investigate the sets of all the bi-invariant connections, using the relation (3.1). We fix a basis {Ei | i = 1, n} in L(G) and denotecijk the structural constants and by Γijkthe coefficients of an arbitrary bi-invariant connection onG. Then (5.1) leads to the linear system
Γsijcksh= Γkiscsjh+ Γksjcsih,
for everyi, j, k, h= 1, n, with (at first sight)n3 unknowns Γijk andn4 equations.
(iv) For a commutativeG, the setsC(G)b andC(G)l coincide, so, in the sequel of this paragraph we shall supposeGnon-commutative.
Remark 5.3. Consider now a non-commutative 2-dimensional Lie group G as in Remark 2.4. We make the convention that the vanishing coefficients of the connections are not written anymore. A short calculation shows that:
(i) the bi-invariant connections verify
∇E1E2=aE1 , ∇E2E1=bE1 , ∇E2E2= (a+b)E2,
for every a, b ∈ R. For a = b = 0, a = b = −1 and a = b = −12 we find the Cartan-Schouten connections∇−,∇+ and∇0respectively.
(ii) the symmetric bi-invariant connections verify
∇E1E2= (b+ 1)E1 , ∇E2E1=bE1 , ∇E2E2= (2b+ 1)E2, for everyb∈R. Forb=−12 we find the Cartan-Schouten connection ∇0.
(iii) the flat bi-invariant connections verify
∇E2E1=bE1 , ∇E2E2=bE2
for everyb∈R. Forb= 0 we find the Cartan-Schouten connection∇−.
(iv) there exists a unique flat and symmetric bi-invariant connection, given by
∇E2E1=−E1 , ∇E2E2=−E2.
To our knowledge, this remarkable bi-invariant connection is a new one! We call itthe Euclidean connection of the 2-dimensional non-commutative Lie groups. This connection cannot be a Levi-Civita connection of a bi-invariant metric onG, because it would imply thatGis compact, and thus (due to the dimension 2) commutative.
One can deduce some simple properties of the differential affine manifold (G,∇):
the auto-parallel left invariant vector fields X ∈ L(G) (i.e. with ∇XX = 0) are exactly those collinear withE1; there exists no parallel left-invariant vector fieldsY (i.e. with∇ZY = 0, for everyZ∈L(G)).
Consider a realization ofG, as the so-called ”ax+b”-group, i.e. the affine group of transformations of the real line. We may identify it with the productR∗×R, with coordinates (x1, x2) and with the multiplication
(a1, a2)(b1, b2) = (a1b1, a1b2+a2).
The Lie algebra of G admits a basis {E1, E2}, with E1 = x1∂2 and E2 = −x1∂1, such that [E1, E2] = E1. In the (global) coordinates (x1, x2), the components of∇ all vanish, which proves that the bi-invariant connection discovered above is exactly the canonical ”Euclidean” connection induced onG. Its auto-parallel curves are the real lines of the plane, restricted to G; we remark that there exist non-complete auto-parallel curves (those which cannot pass through the origin).
Example 5.4. Consider now the non-commutative 3-dimensional Lie group H3 as in Remark 2.6. A tedious calculation shows that, on H3, we have the following properties.
(i) The set of bi-invariant connections is given by all the real numbers Γijk∈R27, withi, j, k= 1,3, such that Γ311,Γ312,Γ321,Γ322 are arbitrary and
Γ122= Γ123= Γ133= Γ132= Γ131= Γ113= Γ211= Γ213= Γ231= Γ233= Γ232= Γ223= Γ333= 0 Γ212= Γ313, Γ111= Γ313+ Γ331, Γ222= Γ323+ Γ332, Γ112= Γ332, Γ221= Γ331, Γ121= Γ323. This set may be parameterized as a product V ×R4, where V is a 4-dimensional subspace in R10 (determined by the last previous relation). Hence, the set of bi- invariant connections may be modelled as a 8-dimensional subspace inR14.
(ii) The set of the symmetric bi-invariant connections is modelled by a 6-dimensional affine subspace inR14, of the formW×R2, whereW is a 4-dimensional affine subspace inR12, defined by
Γ121= Γ112, Γ221= Γ212, Γ331= Γ313, Γ332= Γ323, Γ212= Γ313, Γ111= 2Γ313, Γ222= 2Γ323, Γ112= Γ323, Γ321= Γ312−1 and arbitrary Γ311,Γ322. (The null components are the same as in (i)).
(iii) Similar computations may be made for the bi-invariant connections which are: flat, flat and symmetric, Ricci-flat, Ricci-flat and symmetric, Ricci-symmetric, Ricci-symmetric and symmetric, respectively.
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Authors’ address:
Gabriel-Teodor Pripoae
Department of Mathematics,University of Bucharest, Str. Academiei 14, Bucharest, Romania.
E-mail: gpripoae@fmi.unibuc.ro Cristina-Liliana Pripoae
Department of Applied Mathematics,Academy of Economic Studies, Piata Romana 6, Bucharest, Romania.
E-mail: cristinapripoae@csie.ase.ro
